To determine which of the given options correctly represents the system of equations derived from the pricing of the vegetables, let's examine the information provided:
1. The prices are given for packages containing specified amounts of tomatoes (a), eggplants (b), and potatoes (c):
- 2 tomatoes, 1 eggplant, and 3 potatoes for $10
- 3 tomatoes, 2 eggplants, and 4 potatoes for $14
- 4 tomatoes, 2 eggplants, and 6 potatoes for $20
We can express these scenarios as a system of linear equations:
[tex]\[
\begin{cases}
2a + 1b + 3c = 10 \\
3a + 2b + 4c = 14 \\
4a + 2b + 6c = 20
\end{cases}
\][/tex]
Now, let's compare each option to see if it matches these equations:
Option A:
[tex]\[
\begin{cases}
2a + b + 3c = 10 \\
3a + 2b + 4c = 14 \\
4a + 2b + 6c = 20
\end{cases}
\][/tex]
This option exactly matches the equations we derived.
Option B:
[tex]\[
\begin{cases}
2a + 2b + 3c = 10 \\
3a + 2b + 4c = 14 \\
2a + 3b + 6c = 20
\end{cases}
\][/tex]
The first equation here is incorrect because the correct equation from the problem is \(2a + b + 3c = 10\), not \(2a + 2b + 3c = 10\). Therefore, this option does not match.
Option C:
[tex]\[
\begin{cases}
2a + 2b + 2c = 10 \\
3a + 2b + 4c = 14 \\
4a + 2b + 6c = 20
\end{cases}
\][/tex]
Again, the first equation is incorrect because the correct equation is \(2a + b + 3c = 10\), not \(2a + 2b + 2c = 10\). Therefore, this option does not match.
Option D:
[tex]\[
\begin{cases}
a + 2b + 4c = 14 \\
2a + 3b + 6c = 20
\end{cases}
\][/tex]
The equations provided in this option do not match any of the original given equations. Hence, this option also does not match.
Thus, the system of equations that best represents the given situation is:
[tex]\[
\boxed{\text{Option A}}
\][/tex]
